D-branes and T-duality

We show how the T-duality between D-branes is realized (i) on p-brane solutions (p = 0,…,9) of IIA/IIB supergravity and (ii) on the D-brane actions (p = 0,…, 3) that act as source terms for the p-brane solutions. We point out that the presence of a cosmological constant in the IIA theory leads, by the requirement of gauge invariance, to a topological mass term for the worldvolume gauge field in the 2-brane case.     

Poisson-Lie T-duality and complex geometry in N = 2 superconformal WZNW models

Poisson-Lie T-duality in N = 2 superconformal WZNW models on the real Lie groups is considered. It is shown that Poisson-Lie T-duality is governed by the complexifications of the corresponding real groups endowed with Semenov-Tian-Shansky symplectic forms, i.e. Heisenberg doubles. Complex Heisenberg doubles are used to define on the group manifolds of the N = 2 superconformal WZNW models the natural actions of the isotropic complex subgroups forming the doubles. It is proved that with respect to these actions N = 2 superconformal WZNW models admit Poisson-Lie symmetries. The Poisson-Lie T-duality transformation maps each model onto itself but acts non-trivially on the space of classical solutions.     

Isotropic Kähler structures on Engel 4-manifolds

Examples of isotropic Kähler manifolds (i.e., J²=0) which are neither complex nor symplectic, and therefore not indefinite Kähler, are constructed.     

The first eigenvalue of the Dirac operator on Kähler manifolds

If M^2m is a closed Kähler spin manifold of positive scalar curvature R, then each eigenvalue λ of type r (r {1, …, [(m + 1)/2]}) of the Dirac operator D satisfies the inequality λ² ≥ rR₀/4r − 2, where R₀ is the minimum of R on M^2m. Hence, if the complex dimension m is odd (even) we have the estimation

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for the first eigenvalue of D. In the paper is also considered the limiting case of the given inequalities. In the limiting case with m = 2r − 1 the manifold M^2m must be Einstein. The manifolds S², S² × S², S² × T², P³( ), F( ), P³( ) × T² and F( ³) × T², where F( ³) denotes the flag manifold and T² the 2-dimensional flat torus, are examples for which the first eigenvalue of the Dirac operator realizes the limiting case of the corresponding inequality. In general, if M^2m is an example of odd complex dimension m, then M^2m × T² is an example of even complex dimension m + 1. The limiting case is characterized by the fact that here appear eigenspinors of D² which are Kählerian twistor-spinors.     

Kähler gauge dependence of generalized non-linear σ-model with local complex supersymmetry in 1 + 1 dimensions

The 1 + 1 dimensional analogue of the quantizaton procedure of the gravitational constant by Witten and Bagger in a generalized non-linear σ-model is studied. By making use of the tensor calculus in 1 + 1 dimensions, the model is constructed and found to be manifestly independent of the Kähler gauge transformation. Therefore, the topological properties are not produced in the 1 + 1 dimensional models.     

Generalized hyper-Kähler geometry and supersymmetry

We propose the definition of (twisted) generalized hyper-Kähler geometry and its relation to supersymmetric non-linear sigma models. We also construct the corresponding twistor space.     

Random geometry, quantum gravity and the Kähler potential

We propose a new method to define theories of random geometries, using an explicit and simple map between metrics and large hermitian matrices. We outline some of the many possible applications of the formalism. For example, a background-independent measure on the space of metrics can be easily constructed from first principles. Our framework suggests the relevance of a new gravitational effective action and we show that it occurs when coupling the massive scalar field to two-dimensional gravity. This yields new types of quantum gravity models generalizing the standard Liouville case.     

Characteristic polynomials of random Hermitian matrices and Duistermaat–Heckman localisation on non-compact Kähler manifolds

We reconsider the problem of calculating a general spectral correlation function containing an arbitrary number of products and ratios of characteristic polynomials for a N×N random matrix taken from the Gaussian Unitary Ensemble (GUE). Deviating from the standard “supersymmetry” approach, we integrate out Grassmann variables at the early stage and circumvent the use of the Hubbard–Stratonovich transformation in the “bosonic” sector. The method, suggested recently by J.V. Fyodorov [Nucl. Phys. B 621 [PM] (2002) 643], is shown to be capable of calculation when reinforced with a generalisation of the Itzykson–Zuber integral to a non-compact integration manifold. We arrive to such a generalisation by discussing the Duistermaat–Heckman localisation principle for integrals over non-compact homogeneous Kähler manifolds. In the limit of large-N the asymptotic expression for the correlation function reproduces the result outlined earlier by A.V. Andreev and B.D. Simons [Phys. Rev. Lett. 75 (1995) 2304].     

Generalized Kähler geometry and the pluriclosed flow

In Streets and Tian (2010) [1] the authors introduced a parabolic flow for pluriclosed metrics, referred to as pluriclosed flow. We also demonstrated in Streets and Tian (2010) (preprint) [2] that this flow, after certain gauge transformations, gives a class of solutions to the renormalization group flow of the nonlinear sigma model with B-field. Using these transformations, we show that our pluriclosed flow preserves generalized Kähler structures in a natural way. Equivalently, when coupled with a nontrivial evolution equation for the two complex structures, the B-field renormalization group flow also preserves generalized Kähler structure. We emphasize that it is crucial to evolve the complex structures in the right way to establish this fact.     

Some remarks on special Kähler geometry

Given a special Kähler manifold M$M$, we give a new, direct proof of the relationship between the quaternionic structure on T^∗M$T^{\ast }M$ and the variation of Hodge structures on T^CM$T^{\mathbb{C}}M$.     

The Selberg zeta function and a new Kähler metric on the moduli space of punctured Riemann surfaces

A local index theorem for families of -operators on Riemann surfaces with functures is proved. A new Kähler metric on the moduli space of punctured surfaces is described in terms of the Eisenstein-Maass series.     

Quaternionic and hyper-Kähler metrics from generalized sigma models

The problem of finding new metrics of interest, in the context of SUGRA, is reduced to two stages: first, solving a generalized BPS sigma model with full quaternionic structure proposed by the authors and, second, constructing the hyper-Kähler metric, or suitable deformations of this condition, taking advantage of the correspondence between the quaternionic left-regular potential and the hyper-Kähler metric of the target space. As illustration, new solutions are obtained using generalized Q-sigma model for Wess–Zumino type superpotentials. Explicit solutions analog to the Berger?s sphere and Abraham–Townsend type are given and generalizations of 4-dimensional quaternionic metrics, product of complex ones, are shown and discussed.     

Nahm's equations and hyperkähler geometry

The geometry of certain moduli spaces of solutions to Nahm's equations is studied, and a family of gravitational instationsj is shown to arise as a deformation of theh Atiyah-Hitchin maniford.Address from 1st October 1993: Max-Planck-Institut für Mathematik, Gottfried-Claren-Strasse 26 D-53225 Bonn, Germany